Proposition 1

Let Q and Q' be two quadratic forms over A, non-degenerate and of finite dimensions. In order that Q and Q' be equivalent, it is necessary and sufficient that they have the same dimension and the same type.

The condition is obviously necessary. Suppose that it is satisfied. There then exist neutral forms N, N' such that Q ⊕ N and Q' ⊕ N' are equivalent. Since these two forms have the same dimension, the same is true of N and N', which are consequently equivalent (§ 4, no 2, cor. 2 of prop. 2). Hence Q and Q' are equivalent by virtue of Witt's theorem (§ 4, no 3, cor. 1 of th. 1).